The Wolfram atom

Here we show how we get to the converged ground state of the Wolfram atom (Z=74). It is the open-shell system with four unpaired electrons in the 5d shell:

W: [Xe] 4f(14) 5d(4) 6s(2)

In the following tutorial, for simplicity, we use the two-component Hamiltonian (.X2C) and the smallest available basis set by K.Dyall, v2z. Due to the covnergence problem of the standalone AMFI atomic SCF code we keep the +4 charge (.AMFICH) of W for mean-field orbitals.

Starting with \(W^{4+}\)

Based on experience with d open-shell systems, the first step is to obtain converged state of the (closest) closed-shell cation, \(W^{4+}\). We do not lean on the program’s auto-occupation scheme, rather we ensure proper spinors occupation of two fermion irreducible representations:

$PAM --noarch --mw=300 --mol=W.dyall_v2z.lsym.mol --inp=W_4+.x2c.scf_30_40.2fs.inp  --get "DFCOEF=DFCOEF.W_4+"

(Related files to download are W.dyall_v2z.lsym.mol and W_4+.x2c.scf_30_40.2fs.inp )

The total SCF energy of this cation is -16146.075957236933 a.u., and, as a sign of correctness of the SCF calculation, there are properly degenerated p, d etc eigenvalues, visible in the analysis output (see the full output file W_4+.x2c.scf_30_40.2fs_W.dyall_v2z.lsym.out). In the case of ill SCF process, though converged, atomic shells are not degenerated as they should be.

The W atom from \(W^{4+}\)

In the next step we reuse saved molecular spinors from file DFCOEF.W_4+ and keep the overlap selection

$PAM --noarch --mw=300 --mol=W.dyall_v2z.lsym.mol --inp=W.x2c.scf_ovlsel.2fs.inp --put "DFCOEF.W_4+=DFCOEF"

(File to download W.x2c.scf_ovlsel.2fs.inp)

One has to take the dynamic overlap selection as the mere \(M_{J}\) selection does not work (you can try it for yourself with the input file W.x2c.scf_Mjsel.inp).

For the W atom, we get the total SCF energy of -16149.950403420618 a.u. and, for example, correct occupation of the 5d shell, which lies above the 6s doubly occupied level

                           1/2   3/2  5/2
open shell #1 (f=0.4000):    2    2    1

This can be easily verified as the \(d_{3/2}\) shell gives values of \(M_{J}=\frac{1}{2},\frac{3}{2}\) and the \(d_{5/2}\) spans values of \(M_{J}=\frac{1}{2},\frac{3}{2},\frac{5}{2}\), what makes the 5d open-shell occupation, 2 2 1, as shown above. See also the complete output file, W.x2c.scf_ovlsel.2fs_W.dyall_v2z.lsym.out

How about lower symmetry ?

We descend to the \(C_{2v}\) symmetry, having 1 fermion irreducible representation. Again, we start from the closed shell cation, \(W^{4+}\), and use its DFCOEF for the next run. First, we discover that the \(W^{4+}\) system does not have proper degeneracy of (l>0) spinors in the \(C_{2v}\) symmetry, though the \(W^{4+}\) system is converging. For the subsequent, spinors reordering run, one has to place restrictions on the order of MO and get its DFCOEF saved.

Unfortunately, subsequent calculations in the C2v symmetry do not lead to properly ordered and proprely degenerated orbitals of \(W^{4+}\), therefore (at least we assume), no success can be achieved with the neutral W atom. Maybe some more test with various pre-SCF reordering of starting \(W^{4+}\) spinors would fix this issue ?

(Input files to download for investigating the \(C_{2v}\) symmetry are W.dyall_v2z.C2v.mol, W_4+.x2c.scf.1fs.inp, W.x2c.scf_ovlsel.1fs.inp, W_4+.x2c.scf_reorder_ovlsel.1fs.inp)